Mathematics http://hdl.handle.net/10311/62 2023-05-29T23:05:51Z 2023-05-29T23:05:51Z Some relations between admissible monomials for the polynomial algebra Mothebe, Mbakiso Fix Uys, Lafras http://hdl.handle.net/10311/1966 2019-07-30T00:00:30Z 2015-01-01T00:00:00Z Some relations between admissible monomials for the polynomial algebra Mothebe, Mbakiso Fix; Uys, Lafras Let P(𝑛) = F2[𝑥1, . . . , 𝑥𝑛] be the polynomial algebra in 𝑛 variables 𝑥𝑖, of degree one, over the field F2 of two elements. The mod-2 Steenrod algebra A acts on P(𝑛) according to well known rules. A major problem in algebraic topology is of determining A+P(𝑛), the image of the action of the positively graded part of A. We are interested in the related problem of determining a basis for the quotient vector space Q(𝑛) = P(𝑛)/A+P(𝑛). Q(𝑛) has been explicitly calculated for 𝑛 = 1, 2, 3, 4 but problems remain for 𝑛 ≥ 5. Both P(𝑛) = ⨁𝑑≥0P𝑑(𝑛) and Q(𝑛) are graded, where P𝑑(𝑛) denotes the set of homogeneous polynomials of degree 𝑑. In this paper, we show that if𝑢 = 𝑥𝑚1 1 ⋅ ⋅ ⋅ 𝑥𝑚𝑛−1 𝑛−1 ∈ P𝑑 󸀠 (𝑛−1) is an admissible monomial (i.e., 𝑢 meets a criterion to be in a certain basis forQ(𝑛−1)), then, for any pair of integers (𝑗, 𝜆), 1 ≤ 𝑗 ≤ 𝑛, and 𝜆 ≥ 0, the monomial ℎ𝜆𝑗 (𝑢) = 𝑥𝑚1 1 ⋅ ⋅ ⋅ 𝑥 𝑚𝑗−1 𝑗−1 𝑥2 𝜆−1 𝑗 𝑥𝑚𝑗 𝑗+1 ⋅ ⋅ ⋅ 𝑥𝑚𝑛−1 𝑛 ∈ P𝑑 󸀠 +(2𝜆 −1)(𝑛) is admissible. As an application we consider a few cases when 𝑛 = 5. Some symbols on the abstract may not appear as they appear on the original article. 2015-01-01T00:00:00Z A unified approach to integration theory Robdera, Mangatiana A. http://hdl.handle.net/10311/1879 2019-02-06T00:01:23Z 2019-01-01T00:00:00Z A unified approach to integration theory Robdera, Mangatiana A. We investigate the common features and the resemblance of the central parts of the different existing integration theories to obtain a more unified approach to the notion of integral. Our approach gives a presentation of the integral that does not require the development of measure theory. 2019-01-01T00:00:00Z On the Riesz representation theorem and integral operators Robdera, Mangatiana A. http://hdl.handle.net/10311/1770 2017-11-25T00:01:17Z 2015-12-14T00:00:00Z On the Riesz representation theorem and integral operators Robdera, Mangatiana A. We present a Riesz representation theorem in the setting of extended integration theory as introduced in . The result is used to obtain boundedness theorems for integral operators in the more general setting of spaces of vector valued extended integrable functions. 2015-12-14T00:00:00Z Fundamental Theorem of Calculus in Topological Vector Spaces Robdera, M. A. Kagiso, D. N. http://hdl.handle.net/10311/1766 2017-11-24T00:01:32Z 2017-10-01T00:00:00Z Fundamental Theorem of Calculus in Topological Vector Spaces Robdera, M. A.; Kagiso, D. N. We extend the notions of integration and differentiation to cover the class of functions taking values in topological vector spaces. We give versions of the Lebesgue-Nikodym Theorem and the Fundamental Theorem of Calculus in such a more general setting. 2017-10-01T00:00:00Z